Dozadic number
In mathematics, the dozadic number system (or the 10-adic number system) extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. The extension is achieved by an alternative interpretation of the concept of "closeness" or absolute value. In particular, Dozadic numbers are considered to be close when their difference is divisible by a high power of 10: the higher the power, the closer they are. This property enables dozadic numbers to encode congruence information in a way that turns out to have powerful applications in number theory – including, for example, in the famous proof of Fermat's Last Theorem by Andrew Wiles. Introduction This section is an informal introduction to dozadic numbers. The dozadic numbers are generally not used in mathematics: since 10 is not prime or prime power, the dozadics are not a field. More formal constructions and properties are given below. In the standard dozenal representation, almost allThe number of real numbers with terminating dozenal representations is countably infinite, while the number of real numbers without such a representation is uncountably infinite. real numbers do not have a terminating dozenal representation. For example, 1/5 is represented as a non-terminating dozenal as follows : \frac{1}{5}=0.249724972497\ldots. and 1/7 is represented as a non-terminating dozenal as follows : \frac{1}{7}=0.186\mathcal{X}35186\mathcal{X}35\ldots. Informally, non-terminating dozenal are easily understood, because it is clear that a real number can be approximated to any required degree of precision by a terminating dozenal. If two dozenal expansions differ only after the 10th decimal place, they are quite close to one another; and if they differ only after the 20th decimal place, they are even closer. Dozadic numbers (10-adic numbers) use a similar non-terminating expansion, but with a different concept of "closeness". Whereas two dozenal expansions are close to one another if their difference is a large negative power of 10, two dozadic expansions are close if their difference is a large positive power of 10. Thus 4739 and 5739, which differ by 103, are close in the 10-adic world, and 72694473 and 82694473 are even closer, differing by 107. More precisely, a positive rational number can be uniquely expressed as ·10''d''}}, where and are positive integers and , , and 10 are pairwise relatively prime. Let the "absolute value"The so defined function is not really an absolute value, because the requirement of multiplicativity is violated: |2|_{10} = |2 \cdot 10^0|_{10} = \frac1{10^0} and |5|_{10} = |5\cdot 10^0|_{10} = \frac1{10^0} , but |2 \cdot 5|_{10} = |10^1|_{10} = \frac1{10^1} \ne \frac1{10^0} = |2|_{10} \cdot |5|_{10} . It is, however, good enough for establishing a metric, because this does not need multiplicativity. of 10^d be : |10^d|_{10} := \frac {1} {10^d} . Additionally, we define : |0|_{10} := 0 . Now, taking and we have : , , , with the consequence that we have : \lim_{d \rightarrow +\infty} |10^d|_{10} = 0 . Closeness in any number system is defined by a metric. Using the 10-adic metric the distance between numbers and is given by  −  |10}}. An interesting consequence of the 10-adic metric (or of a -adic metric) is that there is no longer a need for the negative sign. (In fact, there is no order relation which is compatible with the ring operations and this metric.) As an example, by examining the following sequence we can see how unsigned 10-adics can get progressively closer and closer to the number −1: : 9=-1+10 so |9-(-1)|_{10} = \frac {1} {10} . : 99=-1+10^2 so |99-(-1)|_{10} = \frac {1} {100} . : 999=-1+10^3 so |999-(-1)|_{10} = \frac {1} {1000} . : 9999=-1+10^4 so |9999-(-1)|_{10} = \frac {1} {10000} . and taking this sequence to its limit, we can deduce the 10-adic expansion of −1 : |\dots 9999 -(-1)|_{10} = 0 , thus : \dots 9999 =-1 , an expansion which clearly is a ten's complement representation. In this notation, 10-adic expansions can be extended indefinitely to the left, in contrast to decimal expansions, which can be extended indefinitely to the right. Note that this is not the only way to write -adic numbers – for alternatives see the ''Notation'' section below. More formally, a 10-adic number can be defined as : \sum_{i=n}^\infty a_i 10^i where each of the is a digit taken from the set {0, 1, … , 9} and the initial index may be positive, negative or 0, but must be finite. From this definition, it is clear that positive integers and positive rational numbers with terminating decimal expansions will have terminating 10-adic expansions that are identical to their decimal expansions. Other numbers may have non-terminating 10-adic expansions. It is possible to define addition, subtraction, and multiplication on 10-adic numbers in a consistent way, so that the 10-adic numbers form a commutative ring. We can create 10-adic expansions for "negative" numbersMore precisely: additively inverted numbers, because there is no order relation in the 10-adics, so there are no numbers less than zero. as follows : -100 = -1 \times 100 = \dots 9999 \times 100 = \dots 9900 : \Rightarrow -35 = -100+65 = \dots 9900 + 65 = \dots 9965 : \Rightarrow -\left(3+\dfrac{1}{2}\right)=\dfrac{-35}{10}= \dfrac{\dots 9965}{10}=\dots 9996.5 and fractions which have non-terminating decimal expansions also have non-terminating 10-adic expansions. For example : \dfrac{10^6-1}{7}=142857; \qquad \dfrac{10^{12}-1}{7}=142857142857; \qquad \dfrac{10^{18}-1}{7}=142857142857142857 : \Rightarrow-\dfrac{1}{7}=\dots 142857142857142857 : \Rightarrow-\dfrac{6}{7}=\dots 142857142857142857 \times 6 = \dots 857142857142857142 : \Rightarrow\dfrac{1}{7} = -\dfrac{6}{7}+1 = \dots 857142857142857143 = \overline{285714}3. Generalizing the last example, we can find a 10-adic expansion with no digits to the right of the decimal point for any rational number such that is co-prime to 10; Euler's theorem guarantees that if is co-prime to 10, then there is an such that is a multiple of . The other rational numbers can be expressed as 10-adic numbers with some digits after the decimal point. As noted above, 10-adic numbers have a major drawback. It is possible to find pairs of non-zero 10-adic numbers (which are not rational, thus having an infinite number of digits) whose product is 0.See Gérard Michon's article atFor n\in\N_0 let x_n := 6^{5^n} and y_n := 5^{2^n} . We have 6^2 \equiv 6 \text{ mod } 10 and 5^2 \equiv 5 \text{ mod } 10 . Now, : \begin{array}{rlll} ( x_{n+2} - x_{n+1} ) &/ \; ( x_{n+1} - x_n ) \\ = ( {x_n}^{5 \cdot 5} - \; \; {x_n}^5 ) &/ \; ( {x_n}^5 - x_n ) &= {x_n}^{4\cdot 5} &+ {x_n}^{4\cdot 4} &+ {x_n}^{4\cdot 3} &+ {x_n}^{4\cdot 2} &+ {x_n}^{4\cdot 1} \\ = (6^{5^{n+2}} - 6^{5^{n+1}}) &/ \; (6^{5^{n+1}} - 6^{5^n}) &= (6^{5^n})^{4\cdot 5} &+ (6^{5^n})^{4\cdot 4} &+ (6^{5^n})^{4\cdot 3} &+ (6^{5^n})^{4\cdot 2} &+ (6^{5^n})^{4\cdot 1} \\ && \equiv \; \; 6 &+ \; \; 6 &+ \; \; 6 &+ \; \; 6 &+ \; \; 6 \\ &&= 5\cdot 6 \\ &&\equiv 0 &&& \text{ mod } 10 , \end{array} so that 10^n divides x_n-x_{n-1} . This means that the sequence x := \lim_{n\to\infty} x_n converges in the ring of 10-adic numbers. Moreover, it is different from 0, namely x \equiv 6\text{ mod } 10 . Similar facts hold for y := \lim_{n\to\infty} y_n \equiv 5\text{ mod } 10 . But the product (the sequence of the pointwise products) x \cdot y = \lim_{n\to\infty} x_n \cdot y_n is divisible by arbitrarily high powers of 10, so that x \cdot y = 0 in the ring of 10-adic numbers. This means that 10-adic numbers do not always have multiplicative inverses i.e. valid reciprocals, which in turn implies that though 10-adic numbers form a ring they do not form a field, a deficiency that makes them much less useful as an analytical tool. Another way of saying this is that the ring of 10-adic numbers is not an integral domain because they contain zero divisors. The reason for this property turns out to be that 10 is a composite number which is not a power of a prime. This problem is simply avoided by using a prime number or a prime power as the base of the number system instead of 10 and indeed for this reason in -adic is usually taken to be prime.